gen-define-wilson-s-theorem-and-give-one-short-worked-example-7d3a

Define Wilson's theorem and give one short worked example.

# Wilson's Theorem ## Definition **Wilson's Theorem** states that a natural number $n > 1$ is prime if and only if: $$(n-1)! \equiv -1 \pmod{n}$$ In other words, $(n-1)! + 1$ is divisible by $n$ exactly when $n$ is prime. ## Worked Example **Verify that 7 is prime using Wilson's theorem.** We need to check if $(7-1)! \equiv -1 \pmod{7}$ Calculate $6!$: $$6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720$$ Now find $720 \pmod{7}$: $$720 = 102 \times 7 + 6 = 102 \times 7 - 1$$ Therefore: $$720 \equiv -1 \pmod{7}$$ Since $(7-1)! \equiv -1 \pmod{7}$, Wilson's theorem confirms that **7 is prime**. ✓